Answer
$ \sqrt[4] 2(cos82.5^\circ+i\ sin82.5^\circ)$, $ \sqrt[4] 2(cos172.5^\circ+i\ sin172.5^\circ)$, $ \sqrt[4] 2(cos262.5^\circ+i\ sin262.5^\circ)$, $ \sqrt[4] 2(cos352.5^\circ+i\ sin352.5^\circ)$.
Work Step by Step
Based on the given conditions, we have:
$\sqrt 3-i=2(cos330^\circ+i\ sin330^\circ)$, $(1+i)^{1/4}=\sqrt[4] 2(cos(\frac{360k+330}{4})^\circ+i\ sin(\frac{360k+330}{4})^\circ)$, $k=0, z_0=\sqrt[4] 2(cos82.5^\circ+i\ sin82.5^\circ)$, $k=1, z_1=\sqrt[4] 2(cos172.5^\circ+i\ sin172.5^\circ)$, $k=2, z_2=\sqrt[4] 2(cos262.5^\circ+i\ sin262.5^\circ)$, $k=3, z_3=\sqrt[4] 2(cos352.5^\circ+i\ sin352.5^\circ)$.
